There have been two major remedies to overcome this drawback.
The first such effort goes back to Holway [1]}, who extended the local Maxwellian into an ellipsoidal Gaussian
to obtain an additional degree of freedom in computing the transport coefficients.
On the other hand, Shakhov [2]} suggested a way to get the correct Prandtl number by multiplying the Maxwellian with
an extra term that adjusts the heat flux while leaving the collision invariant untouched. The price to pay for this adjustment is that the H-theorem holds only when the flow remains close to the fluid regime. Even with such apparent defect of the model, the Shakhov model has been
widely used in various fields of rarefied gas dynamics, since it reproduces satisfactory qualitative features of the Boltzmann dynamics in many important flow problems [3]}, [4]}, [5]}, [6]}, [7]}, [8]}.
Especially, due to the non-triviality of the heat flux compared to that of the original BGK, it is reported that the Shakhov
model works better for non-isothermal flows [7]}, [4]}. To the best of the authors' knowledge, however, the existence of this important kinetic model has never been studied in the mathematical literature, which is the main motivation of the current work.
